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Thirteenth International Conference on Domain Decomposition Methods
Editors:N.Debit,M.Garbey,R.Hoppe,J.P´eriaux,D.Keyes,Y.Kuznetsov c

2001 DDM.org
46 Analysis of a defect correction method for computational
aeroacoustics
G.S.Djambazov,C.-H.Lai
1
,K.A.Pericleous and Z.-K.Wang
2
Introduction
Many problems of fundamental and practical importance are of multi-scale nature.As a
typical example,the velocity field in turbulent transport problems fluctuates randomly and
contains many scales depending on the Reynolds number of the flow.In another typical
example,which is the main concern of this paper,sound waves are several orders of magnitude
smaller than the pressure variations in the flow field that account for flow acceleration.These
sound waves are manifested as pressure fluctuations which propagate at the speed of sound
in the medium,not as a transported fluid quantity.As a result,numerical solutions of the
Navier-Stokes equations which describe fluid motion do not resolve the small scale pressure
fluctuations.The direct numerical simulation to include the above multiple scale problems is
still an expensive tool for sound analysis [1].
In essence,there are at least three different scales embedded in the flow variables,namely (i)
the mean flow,(ii) flow perturbations or aerodynamic sources of sound,and (iii) the acoustic
perturbation.While flow perturbation or aerodynamic sources of sound may be easier to re-
cover,it is not true for the acoustic perturbation because of its comparatively small magnitude.
¿Froman engineering perspective,much of the larger scales behaviour may be resolved with
the state-of-the-art CFD packages which implement various numerical methods of solving
Navier-Stokes equations.This paper examines,in more detail,a defect correction method,
first proposed in [2],for the recovery of smaller scales that have been left behind.The authors
have demonstrated the accurate computation of (i) and (ii) in [3][4][5].In the present study,
a two-scale decomposition of flow variables is considered,i.e.the flow variable

is written
as


,where


denotes the mean flow and part of aerodynamic sources of sound and

denotes the remaining part of the aerodynamic sources of sound and the acoustic perturbation.
The concept of defect correction [6] has been used in various contexts since the early days.
A typical example of defect correction is the computation of a refined approximation to the
approximate solution


of the nonlinear equation


.Since


is an approximate
solution,the defect may be computed as


.The idea of a defect correction method
is to use a modified/derived version of the original problem such as the one defined by











.If one replaced



as

,then

is the correction
computed by solving






and a refined approximation can be evaluated by using




.More details in expanding the concept to discretised problems and multigrid
methods can be found in [6].Here,the authors would like to concentrate on using the defect
correction concept at the level of the physical problemrather than the discretised problem.For
a given mathematical problemand a given approximate solution,the residue or defect may be
treated as a quantity to measure howwell the problemhas been solved.Such information may
1
C.H.Lai@gre.ac.uk
2
Same address for all authors:School of Computing and Mathematical Sciences,University of Greenwich,30
Park Row,Greenwich,London SE10 9LS,UK
446 DJAMBAZOV,LAI,PERICLEOUS,WANG
then be used in a modified/derived version of the original mathematical problem to provide
an appropriate correction quantity.The correction can then be applied to correct the approx-
imate solution in order to obtain a refined approximate solution to the original mathematical
problem.
This paper follows the basic principle of the defect correction as discussed above and applies
it to the recovery of the propagating acoustic perturbation.The method relies on the use of a
lower order partial differential equation defined on the same computational domain where a
residue exists such that the acoustic perturbation may be retrieved through a properly defined
coarse mesh.
This paper is organised as follows.First,the derivation of a lower order partial differential
equation resulting from the Navier-Stokes equations is given.Truncation errors due to the
model reduction are examined.Second,accurate representation of residue on a coarse mesh
is discussed.The coarse mesh is designed in such a way as to allow various frequencies of
noise to be studied.Suitable interpolation operators are studied for the two different meshes.
Third,numerical tests are performed for different mesh parameters to illustrate the concept.
Finally,future work is discussed.
The defect correction method
The aimhere is to solve the non-linear equation

 



  
 
    
(1)
where


is a non-linear operator depending on

.For simplicity,

is considered to have
two different scales of magnitudes as

 
.Here


is the mean flow and

is the acoustic
perturbation as described in Section 46.Note that



and that







with

much larger than any significant period of the perturbation velocity.The problemhere
is thus purely related to the scale of magnitude.In the case of sound generated by the motion
of fluid,it is natural to imagine

as the Navier-Stokes operator.For a 2-D problem,

 





  




where

is the density of fluid and
 
and
 
are the velocity components along the two spatial
axes.Using the summation notation of subscripts,the 2-DNavier-Stokes problem



  
is written as













 























 
where

is the pressure and


 


 
is the viscous force along

-th axis.
DEFECTCORRECTIONMETHODFORCOMPUTATIONALAEROACOUSTICS 447
Suppose (1) may be split and re-written as


  
 
  



 
 
 



 

 
(2)
where




and
 



are operators depending on the knowledge of


and


  
is a
functional depending on the knowledge of both


and

.Following the concept of defect
correction,


may be considered as an approximate solution to (1).Hence one can evaluate
the residue of (1) as




  
 
 




 





 

which may then be substituted into (2) to give
 



 

  

(3)
In many cases,


  
is small and can then be neglected.In those cases,the problem in
(3) is a linear problem and may be solved more easily to obtain the acoustics fluctuation

.
A non-linear iterative solver is required in order to obtain

for cases when


  
is not
negligible.Finally,to obtain the approximate solution


,one only needs to solve



 

.
Expanding



 
    
for

being the Navier-Stokes operator and re-arranging we
obtain















 






 
   





  

 
























 




and






 

 







 



 





(4)





  

 




  




  

 
 

  

 








 



 

 













 


 



It can be seen that (4) may be written in the formof (3) where
 



 
 























 





 
(5)


  








 


 

 







 

  



 




 


 


 

  



(6)















 





 
 













 























 

(7)
¿Fromthe knowledge of physics of fluids,the acoustic perturbations

and
 
are of very small
magnitude (this is not true for their derivatives),therefore,

may be considered negligible
due to the reason that any feedback fromthe propagatingwaves to the flowmay be completely
ignored,except in some cases of acoustic resonance,which we are not concerned with here.
448 DJAMBAZOV,LAI,PERICLEOUS,WANG
Hence the equation
 



 

,with

given by (5),which is known as the linearised Euler
equation,can be solved in an easier way.The numerics and the techniques involved here are
often referred to as Computational AeroAcoustics (CAA) methods.
The remaining question is to obtain the approximate solution


to the original problem (2).
It is well known that CFD analysis packages provide excellent methods for the solution of



 
 
.Therefore one requires to use a Reynolds averaged Navier-Stokes package
supplemented with turbulence models such as [7,8] to provide a solution of


.One requires


to be as accurate as possible to capture all the physics of interest,such as flow turbulence
and the presence of vortices.
The use of a CFD analysis package effectively solves



 
 
instead of


  
 
 
  
.Following the concept of truncation error in a finite difference method,the truncation
error due to the removal of the perturbation part of the flow variable may be defined by
 


 
 
 




 
 
(8)
Using the relation



 
   



 
 
 




,the truncation error in the present context
is thus given by
  

  
(9)
Note that this truncation error is not related to the discretisation of continuous model.
A two-level multigrid method
In order to simulate accurately the approximate solution,


,to the original problem,



,
the QUICK differencing scheme [9] is used which produces sufficiently accurate results of


for the purpose of evaluating the residue as defined in (7).A sufficiently fine mesh has to
be used in order to preserve vorticity motion.However,much coarser mesh may be used for
the numerical solutions of linearised Euler equations [3,4,5].It certainly has to obey the
Courant limit and also to account for the fact that the acoustic wavelength may be larger than
a typical flow feature which needs to be resolved,e.g.a travelling vortex [10].The present
defect correction method requires to calculate the residue on the CFD mesh and to transfer
these residuals onto the acoustic mesh.Physically,the residue is effectively the sound source
that would have disappeared without the proper retrieval technique as discussed in this paper.
Let

denote the mesh to be used in the Reynolds averaged Navier-Stokes solver.Instead of
evaluating


,one would solve the discretised approximation




 
to obtain



.The
residue on the fine mesh

can be computed as




by means of a higher order approxima-
tion [5].Let

denote the mesh for the linearised Euler equations solver.Again instead of
evaluating

,one would solve the discretised approximation
 








to obtain


.Here
 
is the projection of

onto the mesh

.Let

 
be a restriction operator
to restrict the residue computed on the fine mesh

to the coarser mesh

.The restricted
residue can then be used in the numerical solutions of linearised Euler equations.Therefore
the two-level numerical scheme is (for non-resonance problems):
Solve








  








 



Solve
  







 


 





DEFECTCORRECTIONMETHODFORCOMPUTATIONALAEROACOUSTICS 449
Here


denotes the discretised approximation of the resultant solution on mesh

.Note
that
 
cannot be computed as


 



because

is a non-linear operator.
In the actual implementation,a pressure-density relation which also defines the speed of sound

in air is used:
 




 




(10)
and the first component of the linearised Euler equations in (5) becomes
 



 
 








 














 










 




(11)
The purpose of this substitution is to make sure that the new fluctuations

and
 
do not
contain a hydrodynamic component,and hence can be resolved on regular Cartesian meshes
[4] which is essential for the accurate representation of the acoustic waves or the fluctuation
quantity

.On the other hand,an unstructured mesh may be used to obtain



.The two differ-
ent meshes overlap one another on the computational domain.The computational domain for
the linearised Euler equations is not necessarily the same as the one for the CFD solutions.It
must be large enough to contain at least the longest wavelength of a particular problemunder
consideration or a number of wavelengths where propagation is of interest.The numerical ex-
ample as shown in Section 46 does not contain any complicating solid objects,the restriction
operator
 
 
may then be chosen as an arithmetic averaging process [10].
Numerical experiments with various grid parameters
The propagation of the following one-dimensional pulse is considered:an initial pressure
distribution with a peak in the origin generates two opposite acoustic waves in both directions.
The exact solution of this problem(12) can be verified by substitution in the linearised Euler
equations.







 

 












 
 



 







  


 


(12)
Here

is the amplitude and

is the wavelength of the two sound waves that start from the
origin (
 
) at

 
.The example was reported in [2].This paper provides a detailed
numerical study on various aspects of the grid parameters being used in the two-level method.
The CFD domain is of 12 wavelengths and the CAA domain is of 14 wavelengths.
The effects of the following parameters on the solution accuracyare studied.These parameters
are (a) the ratio H:h,(b) number of points per wavelength,and (c) the restriction operator for
residual transfer fromfine grid to coarse grid.In all cases,the norm






is compared.
Here


is the approximation obtained on the coarse mesh (CAA) after correction and

is
the exact solution of the pressure variable.
Let
 
and
 
be the step lengths in the temporal axis for the CFDmesh and the CAAmesh
respectively.Figure 1 shows the effect on the accuracy for Case (a).Here
 
and
 
are
450 DJAMBAZOV,LAI,PERICLEOUS,WANG
0
2
4
6
8
10
12
14
16
18
20
0
1
2
3
4
5
6
7
8
9
10
11
||P_H - P||_infty
Propagation distance (wavelengths)
h = 0.05, dt_H = 0.00005875, dt_h = 0.000235
H / h = 1
H / h = 2
H / h = 4
H / h = 8
0
2
4
6
8
10
12
14
0
1
2
3
4
5
6
7
8
9
10
11
||P_H - P||_infty
Propagation distance (wavelengths)
h = 0.025, dt_H = 0.00005875, dt_h = 0.000235
H / h = 1
H / h = 2
H / h = 4
H / h = 8
Figure 1:The effect of mesh ratio

:

on the accuracy.
0
5
10
15
20
25
0
1
2
3
4
5
6
7
8
9
10
11
||P_H - P||_infty
Propagation distance (wavelengths)
H = h = 0.00005875, dt_h = 0.000235
5 points
8 points
12 points
16 points
20 points
Figure 2:The effect of number of grid points per wavelength on the accuracy.
chosen to be 0.000235 and 0.00005875 respectively.Two different mesh sizes for the CFD
are chosen and they are 0.05 and 0.025.It can be seen that when

is not fine enough,say

= 0.05,to resolve some of the physics,it is still possible to use the mesh

 

or



to recover the small scale signal.If a finer mesh was used,say

 



,it is possible to
use

 
.This property essentially links with the Courant number of the coarse mesh for
CAA [5],i.e.H,and is also confirmed in the test performed for Case (b).
Figure 2 shows the effect on the accuracy for Case (b).The most accurate solution may be
achieved with more than 12 grid points per wavelength,e.g.16 or more grid points.This
confirms the theoretical study based on Courant limits as discussed in [5].For number of grid
points per wavelength less than 12,the accuracy deteriorates very fast.
Figure 3 shows the effect on the accuracy for Case (c).The restriction operators being used
in this test to transfer the function


onto the coarse mesh

includes
3 point formula:
 


 





 
 
 

   

5 point formula:
 
   






 
 
 

 
 
 



 



7 point formula:
 
   






 
 
 

  


 

 


 
 



 



9 point formula:
 
   




  

 
 

 


  


 


 



 


 
 



 



DEFECTCORRECTIONMETHODFORCOMPUTATIONALAEROACOUSTICS 451
1.7
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1
0
1
2
3
4
5
6
7
8
9
10
11
||P_H - P||_infty
Propagation distance (wavelengths)
H / h = 4, h = 0.015625, dt_H = 0.00005875, dt_h = 0.000235
3 point restriction
5 point restriction
7 point restriction
9 point restriction
1.75
1.8
1.85
1.9
1.95
2
2.05
2.1
2.15
0
1
2
3
4
5
6
7
8
9
10
11
||P_H - P||_infty
Propagation distance (wavelengths)
H / h = 8, h = 0.0078125, dt_H = 0.00005875, dt_h = 0.000235
3 point restriction
5 point restriction
7 point restriction
9 point restriction
Figure 3:The effect of restriction operators on the accuracy.
For very fine CFD mesh,one can retrieve the small scale signal even on a relatively coarse
mesh.In the present study,with

 

 
 


one can use
 
 
while still maintain-
ing the accuracy.The accuracy exhibited by using the coarse mesh


 
 

 

is
compatible with the result for Case (a) as depicted in Figure 1.
Conclusions
This paper provides a numerical method for the retrieval of sound signals using the defect
correction method.A detailed numerical experiments to examine various grid parameters are
provided.Truncation error of solving



 
  
instead of


  
 
    
is derived.
The authors are currently applying the present method to sound propagation in vortex-vortex
interactions.
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